System and method for model based multi-patterning optimization

ABSTRACT

Some embodiments provide a method for optimally decomposing patterns within particular spatial regions of interest on a particular layer of a design layout for a multi-exposure photolithographic process. Specifically, some embodiments model the spatial region using a mathematical equation in terms of two or more intensities. Some embodiments then optimize the model across a set of feasible intensities. The optimization yields a set of intensities such that the union of the patterns created/printed from each exposure intensity most closely approximates the patterns within the particular regions. Based on the set of intensities, some embodiments then determine a decomposition solution for the patterns that satisfies design constraints of a multi-exposure photolithographic printing process. In this manner, some embodiments achieve an optimal photolithographic printing of the particular regions of interest without performing geometric rule based decomposition.

CLAIM OF BENEFIT TO PRIOR APPLICATIONS

This application is a continuation application of U.S. patentapplication Ser. No. 12/189,692, filed Aug. 11, 2008, now issued as U.S.Pat. No. 8,069,423. U.S. Pat. No. 8,069,423 is incorporated herein byreference.

FIELD OF THE INVENTION

The invention relates to the design and manufacture of integratedcircuits. Specifically, the invention relates to systems and methods forimproving the photolithographic printability of an integrated circuitdesign layout.

BACKGROUND OF THE INVENTION

An integrated circuit (“IC”) is a device (e.g., a semiconductor device)or electronic system that includes many electronic components, such astransistors, resistors, diodes, etc. These components are ofteninterconnected to form multiple circuit components, such as gates,cells, memory units, arithmetic units, controllers, decoders, etc. An ICincludes multiple layers of wiring that interconnect its electronic andcircuit components.

Design engineers design ICs by transforming logical or circuitdescriptions of the ICs' components into geometric descriptions, calleddesign layouts. IC design layouts typically include (1) circuit modules(i.e., geometric representations of electronic or circuit IC components)with pins, and (2) interconnect lines (i.e., geometric representationsof wiring) that connect the pins of the circuit modules. A net istypically defined as a collection of pins that need to be connected. Inthis fashion, design layouts often describe the behavioral,architectural, functional, and structural attributes of the IC.

To create the design layouts, design engineers typically use electronicdesign automation (“EDA”) applications. These applications provide setsof computer-based tools for creating, editing, analyzing, and verifyingdesign layouts.

Fabrication foundries (“fabs”) manufacture ICs based on the designlayouts using a photolithographic process. Photolithography is anoptical printing and fabrication process by which patterns on aphotolithographic mask (i.e., photomask) are imaged and defined onto aphotosensitive layer coating a substrate. To fabricate an IC, photomasksare created using the IC design layout as a template. The photomaskscontain the various geometries (i.e., features) of the IC design layout.The various geometries contained on the photomasks correspond to thevarious base physical IC elements that comprise functional circuitcomponents such as transistors, interconnect wiring, via pads, as wellas other elements that are not functional circuit elements but that areused to facilitate, enhance, or track various manufacturing processes.Through sequential use of the various photomasks corresponding to agiven IC in an IC fabrication process, a large number of material layersof various shapes and thicknesses with various conductive and insulatingproperties may be built up to form the overall IC and the circuitswithin the IC design layout.

As circuit complexity continues to increase and transistor designsbecome more advanced and ever smaller in size (i.e., die shrink),advances in photolithographic processes are being pursued to enable thelarge scale, low cost manufacturing of such circuits. However,constraining factors in traditional photolithographic processes limitthe effectiveness of current photolithographic processes. Some suchconstraining factors are the lights/optics used within thephotolithographic processing systems. Specifically, the lights/opticsare band limited due to physical limitations (e.g., wavelength andaperture) of the photolithographic process. Therefore, thephotolithographic process cannot print beyond a certain pitch, distance,and other such physical manufacturing constraints.

A pitch specifies a sum of the width of a feature and the space on oneside of the feature separating that feature from a neighboring feature.Depending on the photolithographic process at issue, factors such asoptics and wavelengths of light or radiation restrict how small thepitch may be made before features can no longer be reliably printed to awafer or mask. As such, the smallest size of any features that can becreated on a wafer is severely limited by the pitch.

FIG. 1 illustrates a typical pitch constraint of a photolithographicprocess. In FIG. 1, a pitch 110 acts to constrain the spacing betweenprintable features 120 and 130 of a design layout. While otherphotolithographic process factors such as the threshold 140 can be usedto narrow the width 150 of the features 120 and 130, such adjustments donot result in increased feature density without adjustments to the pitch110. As a result, increasing feature densities beyond a certainthreshold is infeasible via a pitch constrained single exposure process.

Certain reticle enhancement techniques (RET) allow for photolithographicprocesses to extend beyond some of the various photolithographicmanufacturing constraints such as the width constraint. Some commontechniques include: using optical proximity correction (OPC) to distortphotomask shapes to compensate for image errors resulting fromdiffraction or process effects that cause pattern inaccuracies, usingoff-axis illumination (OAI) for optimizing the angle of illumination fora particular pitch, using alternating phase shift masks (PSM) forimproving lithographic resolution by introducing a particular phaseshift between adjacent patterns or features on a photomask, and usingscatter bars to place narrow lines or spaces adjacent to a feature inorder to make a relatively isolated line behave more like a dense line.

However, these and other techniques are limited both by cost andeffectiveness. To overcome these and other constraints, some fabricationprocesses have implemented a multiple exposure photolithographic processas illustrated in FIG. 2.

In FIG. 2, a design layout 205 specifies three features 210-230 that arepitch constrained and therefore cannot be photolithographically printedwith a conventional single exposure process. Analysis of thecharacteristics (e.g., the band limitation) of the availablephotolithographic process and of the design layout 205 results in thedecomposition of the design layout 205 into a first exposure 240 forprinting features 210 and 230 and a second exposure 250 for printingfeature 220. As such, the features 210 and 230 are assigned to a firstphotomask for printing during the first exposure 240 and feature 220 isassigned to a second photomask for printing during the second exposure250.

FIG. 3 illustrates a decomposition of a pattern 310 defined in a layerof design layout for fabricating an IC into two sets of polygons 320 and330. Each such decomposed set of polygons 320 and 330 is printed duringan exposure of a multiple exposure photolithographic printing process.For instance, polygon set 320 is printed during a first exposure inorder to produce contours 340 and polygon set 330 is printed during asecond exposure in order to produce contours 350. The resulting union ofthe contours 340 and 350 generates pattern 360 that is sufficient toapproximately reproduce the original pattern 310. Accordingly, a validdecomposition solution is such that the union of the contourscreated/printed from each exposure closely approximates specificationswithin the original design layout and satisfies multi-exposurephotolithographic printing constraints (e.g., the band limit and thetarget layout specified within the design layout) with no resulting“opens”, “shorts”, or other printing errors materializing on thephysical wafer.

Existing decomposition tools for decomposing layouts into one or moreexposures are often rule based and proceed on a pattern by pattern(i.e., geometry by geometry) basis. Such tools are therefore applicableonly to patterns for which a pre-programmed or known decompositionsolution exists within a library. These tools are effective fordecomposing simple designs with regular repeating patterns such asgradings including lines and spaces. However, design layouts with moresophisticated geometries (e.g., logic designs and microprocessor designscontaining complex patterns and shapes with bends and jogs) cannot beprocessed using these existing decomposition tools as solutions do notexist for the unique patterns appearing within such design layouts.Instead, layout designers would be notified of the patterns for which aknown solution does not exist. The layout designers would then beprompted to manually produce a decomposition solution for thesepatterns. To remedy this issue, solution providers must continuallyupdate the pattern libraries and algorithms used by these tools toaccommodate the growing complexity of the design layouts.

Moreover, existing decomposition tools are inefficient in the manner bywhich they perform decomposition analysis. Repeated polygonal patternswithin a single design layout are each independently analyzed and asolution is provided for each instance as if each instance is the firstsuch instance. Therefore, the more dense a design layout, the more timeand processing resources needed to process and decompose the designlayout. Also, traditional prior art decomposition tools often operate ina local area by local area basis such that solutions provided to remedyprintability issues appearing within a particular local area may have adetrimental effect to other unprocessed or processed local areas of thelayout.

Therefore, there is a need to efficiently and effectively decompose anentire layout or section of a layout while avoiding the pitfallsassociated with geometric rule based decomposition techniques. Moreover,there is need to generalize such techniques and tools such that theapplicability of these techniques and tools is no longer restricted to aknown or pre-programmed set of solutions.

SUMMARY OF THE INVENTION

Some embodiments provide a method for decomposing a particular patternon a layer of a design layout into multiple patterns that can beproduced in multiple photolithographic operations. Based on a particularmodel, the method searches through the space of lithographicallyfeasible images to identify a set of at least two images for decomposingthe particular pattern. In some embodiments, the particular model is animage-intensity model that bounds the search to only intensity imagesthat are feasible solutions for decomposing the particular pattern. Forinstance, in some embodiments, the model includes two or more Fourierdecomposition equations for two or more intensity images.

Once the method identifies the set of images for decomposing theparticular pattern, the method then determines whether the union of theidentified set of images is sufficiently close to the particularpattern. When the union is not sufficiently similar to the particularpattern, the method repeats the above process by searching again throughthe solution space of feasible photolithographic images to identifyanother set of feasible images that need to be compared with theparticular pattern. When the method identifies a set of images (e.g.,two intensity images) whose union is sufficiently similar to theparticular pattern, the method uses the identified set of images toproduce a set of mask layouts (e.g., two mask layouts). The mask-layoutset can then be used to produce the particular pattern on a substrate oron one of the layers defined above or below a substrate.

BRIEF DESCRIPTION OF THE SEVERAL VIEWS OF THE DRAWINGS

The novel features of the invention are set forth in the appendedclaims. However, for purpose of explanation, several embodiments of theinvention are set forth in the following figures.

FIG. 1 illustrates a typical orthogonal pitch constraint imposed by aphotolithographic process.

FIG. 2 illustrates a fabrication processes implementing a multipleexposure photolithographic process.

FIG. 3 illustrates a decomposition of a pattern into two sets ofpolygons.

FIG. 4 presents a process for decomposing a particular pattern on alayer of a design layout into multiple patterns that can be produced inmultiple photolithographic operations in accordance with someembodiments.

FIG. 5 presents a more-detailed illustration of the process of FIG. 4.

FIG. 6 presents a process for deriving the cost function in accordancewith some embodiments.

FIG. 7 describes a process for imposing a grid on a region of interestto divide the region into a set of nodes.

FIG. 8 illustrates the rounding of edges of a pattern within a region ofinterest in accordance with some embodiments.

FIG. 9 illustrates the demarcation of edge lines for patterns appearingwithin a region of interest using sets of nodes in accordance with someembodiments.

FIG. 10 illustrates defining sets of internal nodes and external nodesthroughout a region of interest in accordance with some embodiments.

FIG. 11 presents a constraint for costing an internal node that is fixedto a particular exposure in accordance with some embodiments.

FIG. 12 illustrates a constraint used to cost an internal node that isnot fixed to any particular exposures in accordance with someembodiments.

FIG. 13 illustrates a constraint used to cost an external node inaccordance with some embodiments.

FIG. 14 illustrates a constraint and corresponding cost function term inaccordance with some embodiments defining two separate thresholds.

FIG. 15 illustrates a constraint imposed on some adjacently definedinternal nodes and its corresponding cost function term in accordancewith some embodiments.

FIG. 16 illustrates a cost function in accordance with some embodiments.

FIG. 17 presents a process for computing a particular cost vector of thecost function in accordance with some embodiments.

FIG. 18 illustrates computing the cost vector for internal nodes thatare fixed to a second exposure in accordance with the process of FIG.17.

FIG. 19 illustrates a Fourier expansion for two separate intensities andin accordance with some embodiments.

FIG. 20 illustrates an example of a region of interest in someembodiments.

FIG. 21 presents a process for identifying optimal intensity images foreach exposure of a multi-exposure photolithographic process inaccordance with some embodiments.

FIG. 22 illustrates a process performed by the polygon generator inaccordance with some embodiments.

FIG. 23 presents a conceptual diagram of the software architecture forimplementing the double patterning optimization method of someembodiments of the invention.

FIG. 24 illustrates a region of interest that has been selected formulti-exposure intensity optimization as encompassing an entire designlayout or layer of the design layout.

FIG. 25 illustrates a region of interest that has been selected formulti-exposure intensity optimization as encompassing only a spatialsub-region of a layer of a design layout.

FIG. 26 is a block diagram of an illustrative computing system suitablefor implementing an embodiment of the present invention.

DETAILED DESCRIPTION OF THE INVENTION

In the following description, numerous details are set forth for purposeof explanation. However, one of ordinary skill in the art will realizethat the invention may be practiced without the use of these specificdetails. In other instances, well-known structures and devices are shownin block diagram form in order not to obscure the description of theinvention with unnecessary detail.

I. Overview

Some embodiments provide a process for decomposing a particular patternon a layer of a design layout into multiple patterns that can beproduced in multiple photolithographic operations for fabricating anintegrated circuit (IC). FIG. 4 illustrates one example of such aprocess for some embodiments of the invention. As shown in FIG. 4, theprocess 400 initially defines (at 410) a model that expresses the rangeof feasible solutions for decomposing the particular pattern. In someembodiments, the model expresses the range of feasible solutionsmathematically. In some embodiments, the particular model is animage-intensity model that bounds the search to only intensity imagesthat are feasible solutions for decomposing the pattern. For instance,in some embodiments, the model includes two or more Fourierdecomposition equations for two or more intensity images.

Next, at 420, the process searches through the solution space oflithographically feasible images to identify a set of at least twoimages. The process then identifies the union (at 430) of the identifiedset of images. The process next determines (at 440) whether theidentified union would produce a pattern that would be sufficientlyclose to the particular pattern that is being decomposed.

When the union is not sufficiently similar to the particular targetpattern, the process 400 returns to 420 to search again through thesolution space of feasible photolithographic images to identify anotherset of feasible images. For this new solution, the process againperforms operations 430 and 440 to determine whether the union of theimages that are part of the new solution is sufficiently close to theparticular target pattern. In some embodiments, the determination isbased on whether the images, when used to construct photomasks for amulti-exposure photolithographic process, will yield lithographicallyprinted features that satisfy printing constraints and sufficientlyrepresent the original target pattern.

Once the process determines (at 440) that the union of a set ofidentified images (e.g., the union of two identified intensity images)is sufficiently similar to the particular pattern, the process defines(at 450) a sub-pattern layout for each identified image. The processthen generates (at 460) a set of photomask layouts (e.g., two photomasklayouts) from the defined sub-pattern layouts. The photomask layouts canthen be used to produce the particular pattern on a substrate or on oneof the layers defined above or below a substrate.

FIG. 5 illustrates a more-detailed example of the operation of someembodiments of the invention. In this example, several operations areperformed to decompose a pattern 505 in a region 510 of a particularlayer of an IC design layout for use in a multi-patterningphotolithographic process. In this example, the pattern 505 isdecomposed into two patterns. However, one of ordinary skill willrealize that the invention can be used to decompose the pattern intothree or more patterns.

As shown in FIG. 5, some embodiments initially impose a grid on theregion to define a set of nodes 520 (shown as circles) in the region.These embodiments then define a cost function F for the region in termsof two intensity values, I₁ and I₂, as described in further detailbelow. To define this cost function F, these embodiments define one ormore cost expressions 525 (e.g., cost expressions C1, C2, etc. in FIG.5) for each node. Each cost expression for a node is based on a costconstraint that is defined in terms of one or both of the intensityvalues, I₁ and I₂, for that node. When a node has more than one costexpression, some embodiments produce a single weighted cost expressionfor the node by using one or more weighting factors to combine theindividual cost expressions of the node.

After defining the cost expression for each node and thereby the costfunction F for the region 510, some embodiments use an optimizer 535that implements an optimization algorithm to search the solution spaceof intensity values to identify an optimal solution for the costfunction. For instance, some embodiments search the solution space tofind a solution that minimizes the cost function F^(T)F, where F isdefined as a function that has a small value when the union of the twointensity images closely matches the original pattern.

To search this solution space, some embodiments define a mathematicalimage-intensity model that bounds the search to only intensity valuesthat are feasible solutions for decomposing the pattern. For instance,in some embodiments, the model for the example illustrated in FIG. 5includes two Fourier decomposition equations for two intensity images.

Accordingly, the output result of the optimization algorithm 535 in someembodiments is a set of Fourier coefficients that can be used to definetwo intensity images 530 a and 530 b (also called intensity graphsbelow) for the region. Each intensity image is a grayscale image. Insome embodiments, a white value in an intensity image represents alocation in the region that should receive light during the exposurecorresponding to the particular intensity, whereas a black value in theintensity image represents a location in the region that should notreceive light during that exposure. A gray value is a value between thewhite and black values; the location of a gray value may or may notreceive light depending on its level of “grayness”.

After obtaining the grayscale intensity images, a pattern generator 550generates two sets of polygons 540 a and 540 b from the intensity images530 a and 530 b. Each polygon set defines a sub-pattern that needs to beproduced during one of two lithographic exposures. In other words, thetwo sets of polygons are the two patterns that decompose the originalpattern 505.

To generate the two sets of polygons 540 a and 540 b (i.e.,sub-patterns), the pattern generator 550 of some embodiments imposes agrid on the intensity images 530 a and 530 b that are the output result,or can be obtained from the output result, of the optimization process535. Some embodiments use a coarser grid than the grid 520 that waspreviously specified to specify the cost function F, although otherembodiments use the same grid or a finer grid on the intensity images.For each node on the imposed grid, some embodiments use a thresholdvalue to convert any node's gray value in an intensity image associatedto a black or white value. These embodiments then generate two polygonlayouts from the two resulting intensity images where the white valuessignify patterns or features that should be lithographically printed(e.g., by using the resulting white values in each intensity image togenerate one or more polygons in the polygon layout).

In some embodiments, the pattern generator 550 generates thesub-patterns by (1) forming the intersection of the polygons in theoriginal pattern 505 with a set of points of the first intensity imagethat exceed the threshold value (i.e., I₁(x,y)>t), and (2) forming theintersection of the polygons in the original pattern 505 with a set ofpoints of the second intensity image that exceed the threshold value(i.e., I₂(x,y)>t). These intersections are approximated by unions ofpolygons. These polygons (1) make up a first pattern 540 a that is usedto make a first photomask, and (2) make up the second pattern 540 b thatis used to make a second photomask.

In the examples above, polygons represent bright features in a darkbackground. This is the case when using a positive photoresist forDamascene metal layers and contact hole layers. However, it should beapparent to one of ordinary skill in the art that some embodiments mightemploy a different optical polarity such that the polygons representdark features in a bright background. In such a case, the pattern 540 awould be derived from the intersection of the original pattern 505 withthe set of points at which I₁(x,y)<t, and pattern 540 b would be derivedfrom the intersection of the original pattern 505 with the set of pointsat which I₁(x,y)<t.

Each polygon layout (i.e., each identified sub-pattern) can then be usedto generate a mask layout for a separate lithograph exposure operationduring the multi-patterning photolithography process that is used tofabricate the IC layout. Any known techniques can be used to generate aparticular photomask from a particular exposure polygon layout. Examplesof some techniques include optical proximity correction andsub-resolution assist features.

The generated photomasks can then be used to generate the desiredexposure sub-patterns to be produced on a desired semiconductorsubstrate layer (i.e., on the substrates itself or on one of the layersabove or below the substrate). This exposure would then create thesub-patterns on the desired semiconductor layer. The union of thecreated sub-patterns would then serve as a close replication 545 of thepattern 505, as shown in FIG. 5.

Several more detailed embodiments of the invention are described in thesections below. Section II describes deriving the cost function based ona set of constraints assigned to nodes defined over a region ofinterest. Next, Section III describes solving the cost function toidentify an optimal set of intensity images based on the set ofconstraints. Section IV provides a software architecture used by someembodiments. Lastly, Section V provides a description of a systemarchitecture with which some embodiments of the invention areimplemented.

II. Cost Function

FIG. 6 presents a process 600 for deriving the cost function inaccordance with some embodiments. In some embodiments, the process 600is performed in conjunction with multi-patterning decomposition. Theprocess begins by selecting (at 610) a layer within a design layout usedfor fabricating an IC. The process determines (at 620) whether a regionof interest on the layer requires decomposition for printing in amulti-exposure photolithographic process. When no such region ofinterest exists on the selected layer, the process determines (at 625)whether additional layers remain. When no layers remain, the processends. Otherwise, the process selects the next region (at 610) andreexamines the layer for any regions of interest.

When a region of interest is identified within a selected layer, theprocess selects (at 630) the region of interest and defines (at 640)nodes over the selected region. The process examines (at 650) one ormore constraints for each particular node in order to define the costfunction. The process then optimizes (at 660) the cost function beforereturning to examine any additional regions of interest within theselected layer. Several more detailed examples for deriving the costfunction are provided below.

A. Node Designation

Some embodiments avoid the geometric pattern by pattern analysisperformed by various other double patterning optimization solutions byimposing a grid over a region of interest in order to divide the regioninto a discretized set of nodes. A sufficient sampling of the region(e.g., a rate equal to or exceeding the Nyquist sampling rate) is takenin order to represent all features within the region of interest withsufficient granularity. In this manner, the continuous space of theregion of interest is mapped to a finite set of parameterizableelements. As such, modifications to the sampling rate affects theaccuracy of the results relative to the processing resources and timeneeded to produce the results.

FIG. 7 describes a process 700 for dividing a region of interest into aset of nodes. The process begins by receiving (at 710) the region ofinterest and processing (720) the patterns (e.g., portions of circuitmodules, interconnect lines, etc.) appearing within the region ofinterest. Specifically, some embodiments process the patterns to createshapes that can achieve more realistic printability based on physicalconstraints (e.g., inability to print perpendicular corner segments) inthe photolithographic printing process. One such processing procedure isto identify corner edges for patterns appearing within the identifiedregion and to round the identified edges. FIG. 8 illustrates therounding of edges 810-840 of a pattern 850 within a region of interest860 in accordance with some embodiments.

Once the processing of the patterns is performed, the process thenimposes a grid on the region by defining a set of nodes as described insteps 730 and 750 below. Different node types are used to demarcatedifferent characteristics of the region represented by each node. Someembodiments utilize a coloring scheme to represent some suchcharacteristics. Moreover, different coloring schemes can be usedthroughout different stages of the double patterning optimization. Forinstance, the description below may refer to some nodes as bright anddark to demarcate intensities for internal nodes from external nodes andlater describe the same nodes using blue and red coloring where each ofthe red and blue colors corresponds to an exposure of a multi-exposurephotolithographic process.

In some embodiments, edge lines for the patterns appearing within theregion of interest are demarcated using a pair of nodes for the purposeof defining a threshold for acceptable edge placement errors.Accordingly, the process defines (at 730) internal target edge nodes andexternal target edge nodes with the distance between the nodesspecifying the acceptable edge placement error. In some embodiments, aninternal node represents a discretized space that falls within thebounds of a pattern or feature appearing within the region of interestand an external node represent a discretized space that falls outsidethe bounds of a pattern or feature. Furthermore, internal edge nodesrepresent the discredited space along the boundary or border of thepattern or feature.

FIG. 9 illustrates the demarcation of edge lines for patterns appearingwithin a region of interest using sets of nodes in accordance with someembodiments. As shown, outer nodes 910 and inner nodes 920 demarcateeach edge of the pattern 930 appearing within the region of interest940.

The process 700 also identifies and defines (at 740) internal nodes andexternal nodes over the remainder of the region of interest. FIG. 10illustrates defining sets of internal nodes (e.g., 1010, 1015, and 1025)and external nodes (e.g., 1020) throughout the region of interest 1040in accordance with some embodiments. In some embodiments, the samplingrate used to determine the number of internal nodes and external nodeshas to equal or exceed the Nyquist sampling rate determined from theNyquist sampling theorem. The Nyquist sampling theorem states that for aband limited signal (e.g., the optics of the photolithographic printingprocess), an equally spaced sampling frequency must exceed the maximumfrequency of the band limited signal by at least a factor of two for thesignal to be reconstructed without aliasing. The sampling rate is thustwice the sample frequency. It should be apparent to one of ordinaryskill in the art that some embodiments modify the sampling rate based onother sampling theorems or based on the resources available to processthe particular region of interest.

Some of these defined internal nodes can further be demarcated fromother internal nodes. In some embodiments, “fixed” internal nodes aredemarcated from “non-fixed” internal nodes. Accordingly, the processidentifies and defines (at 750) fixed node types where possible.

FIG. 10 also illustrates the designation of fixed internal nodes fromnon-fixed internal nodes. As shown, a fixed internal node (e.g., node1010 or 1015) is a node appearing at a boundary of a pattern 1030 withinthe region 1040 where the pattern 1030 is part of a feature that extendsbeyond the particular region of interest (e.g., segments 1050 and 1060).In some embodiments, a node is designated as a fixed node type, becauseportions of the feature that extend beyond the particular region havealready been “fixed” to print with a particular exposure of amulti-exposure printing process. The node will be fixed to print withthe same particular exposure in order to minimize “cuts” to a patternand therefore reduce the number of intensity transitions that occur whenprinting the pattern. Accordingly, the fixed internal node 1010 willalso be fixed to print in the same exposure as the portion 1050extending beyond the particular region of interest 1040. Additionally,designating fixed nodes provide an indication in how to optimize theremainder of unfixed internal nodes defined over the region (e.g.,reduce the number of transitions between exposures for adjacent nodes).

In some embodiments, the internal nodes must be bright and the externalnodes must be dark. This demarcation of internal nodes from externalnodes indicates nodes that require intensity values that exceed a giventhreshold for photolithographic printing. For instance, this demarcationmay be used to define printing of a contact layer or metal layer usingsome positive photoresist. However, it should be apparent to one ofordinary skill in the art that the inverse is similarly applicable tosome embodiments. For instance, some embodiments require external nodesto be bright and internal nodes to be dark when reversing the polarityused for the photolithographic printing.

B. Constraint Assignment

After defining the nodes over the region of interest, some embodimentsdefine one or more design constraints to apply to each node based on thenode type. In some embodiments, the constraints are defined in terms ofintensity values for each exposure of a multi-exposure photolithographicprinting process as described in further detail below. The followingdescription provides an exemplary list of constraints. This listing isnot meant to be an exhaustive listing of constraints. As such, it shouldbe apparent to one of ordinary skill in the art that various otherconstraints may be assigned to some or all such nodes.

Internal nodes represent portions of a pattern within a particularregion of interest. In some embodiments, these nodes requirephotolithographic printing in at least one exposure of a multi-exposurephotolithographic printing process. Accordingly, a constraint isassigned to the internal nodes for each exposure to determine whetherthe node is printed in at least one exposure.

For fixed internal nodes that are fixed to a particular exposure, onlyone such constraint is needed to ensure that the intensity for theparticular exposure is sufficient to print the node. FIG. 11 presents aconstraint for costing an internal node that is fixed to a particularexposure in accordance with some embodiments. A graph pictoriallyrepresents the constraint. As shown, intensity values of the exposureare illustrated across the x-axis of the graph, cost values areillustrated across the y-axis of the graph, and a threshold isillustrated to specify the set of intensity values that incur no costpenalty. Specifically, when the intensity exceeds the threshold, therewill be no cost or a zero cost. For any fixed internal node where theintensity does not exceed the threshold there will be a non-zero costassociated with the node. Such a cost signifies that the printingprocess is unable to correctly print the portion of the patternrepresented by the node. This could result in any number ofphotolithographic errors such as pinching, necking, and various otherprinting errors that often lead to “opens” between otherwise connectedcircuits or “shorts” between otherwise unrelated circuits in thephysical implementation.

A cost function term can be used to express the quantification of thecost constraint illustrated in FIG. 11. In some embodiments, the costfunction term associated with the graph illustrated in FIG. 11 isMax(−I₁(x_(i))+t,0)  (1)where I₁(x_(i)) represents the intensity value at node x_(i) and trepresents the threshold.

As noted above, for internal nodes that are not fixed to a particularexposure, costing is performed in relation to all intensity values forall possible exposures to which the node may be decomposed onto. FIG. 12illustrates a constraint used to cost an internal node that is not fixedto any particular exposures in accordance with some embodiments. Asshown, the photolithographic process is performed in two exposures.Accordingly, a constraint is specified for each exposure. When theintensity defined for either exposure exceeds the threshold needed toprint the node, then the constraint cost for that node is satisfied anda zero value is assigned. In FIG. 12, the x-axis of the graph representsintensity values defined for a first exposure and the y-axis of thegraph represents intensity values defined for a second exposure. Wheneither intensity value exceeds the threshold, a zero cost is assigned tothe node.

However, when neither threshold is exceeded, then the printing processwill not correctly print the portion of the pattern represented by thenode. In some embodiments, the cost function term associated with theconstraint illustrated in the graph of FIG. 12 isMax(−Max(I₁(x_(i)),I₂(x_(i)))+t,0)  (2)where I₁(x_(i)) represents a first intensity value at node x_(i) whenprinting with a first exposure of multi-exposure photolithographicprinting process, I₂(x_(i)) represents a second intensity value at nodex_(i) when printing with a second exposure of multi-exposurephotolithographic printing process, and t represents the threshold.

External nodes represent portions within the particular region ofinterest that should not be printed. Accordingly, a constraint isassigned to the external nodes to ensure that the intensity associatedwith the node does not exceed an intensity threshold for printing theexternal nodes. FIG. 13 illustrates a constraint used to cost anexternal node in accordance with some embodiments. As shown, thephotolithographic process is performed in two exposures (i.e., twointensities). Accordingly, a cost is computed for the external nodebased on each exposure. When the intensity defined for either exposureat the external node exceeds the threshold, then the constraint cost forthat node is violated and a non-zero value is assigned to the node. InFIG. 13, the x-axis of the graph represents intensity values defined fora first exposure and the y-axis of the graph represents intensity valuesdefined for a second exposure. In some embodiments, the cost functionterm associated with the constraint illustrated in the graph of FIG. 13isMax(Max(I₁(x_(i)),I₂(x_(i)))−t,0)  (3)where I₁(x_(i)) represents a first intensity value at node x_(i) whenprinting with a first exposure of multi-exposure photolithographicprinting process, I₂(x_(i)) represents a second intensity value at nodex_(i) when printing with a second exposure of multi-exposurephotolithographic printing process, and t represents the threshold.

It should be apparent to one of ordinary skill in the art that in someembodiments separate thresholds are specified for each intensity value.FIG. 14 illustrates a constraint and corresponding cost function term inaccordance with some embodiments defining two separate thresholds.

FIG. 13 illustrates a constraint used to cost an external node inaccordance with some embodiments. In this figure, t₁ is a largerthreshold than t₂. Accordingly, I₁(x_(i)) must have a higher intensityvalue in order to satisfy t₁ than does I₂(x_(i)) for satisfying t₂. Thecost function term associated with the constraint illustrated in thegraph of FIG. 14 isMax(Max(−I₁(x_(i))+t₁,−I₂(x₁)+t₂),0)  (4)

Any number of additional constraints may be associated to one or morenode types. For instance, some embodiments specify a constraint to placeintensity bounds at each node such that first and second intensityimages must be bounded between values of 0 and 1. The following costfunction terms specify such constraints:Max(−I₁(x_(i)),0)  (5)Max(−I₂(x_(i)),0)  (6)Max(I₁(x_(i))−1,0)  (7)Max(I₂(x_(i))−1,0)  (8)

Some embodiments further assign a constraint to some adjacently definedinternal nodes in order to define a cost when the intensity between theadjacent internal nodes switches. FIG. 15 illustrates one suchconstraint and its corresponding cost function term in accordance withsome embodiments. As shown, the constraint penalizes transitions todifferent exposures for neighboring nodes. The cost function termassociated with this constraint isσ(I₁(x_(i)))−σ(I₁(x_(j)))  (9)where σ is a sigmoid function such that σ(I) is close to a value of 0for intensities I that are less than the threshold t, and σ(I) is closeto a value of 1 for intensities I greater than the threshold t.

C. Defining the Cost Function

From the various constraints assigned to each node, some embodimentsderive an overall cost function in terms of the intensity values. FIG.16 illustrates a cost function 1610 in accordance with some embodiments.Each row within the cost function 1610 represents multiple entries in acost function vector, with one entry for each node of the appropriatetype. It should be apparent to one of ordinary skill in the art thatvarious other cost constraints may be included in addition to or inplace of the enumerated constraints within the illustrated costfunction. Moreover, it should be apparent that not all constraints shownwithin the cost function are applicable to all node types. In fact,certain constraints are exclusive to certain node types.

FIG. 17 presents a process 1700 for computing a particular cost vector(e.g., 1620) of the cost function 1610 in accordance with someembodiments. The process 1700 selects (at 1710) a cost vector. Theprocess identifies (at 1720) one or more node types that are associatedwith the selected cost vector. For instance, node types include internalnodes, fixed internal nodes, external nodes, boundary nodes, brightnodes, darks nodes, etc. Furthermore, each node can belong to one ormore node types.

The process then identifies (at 1730) the nodes (and therefore theircorresponding constraints) associated with the identified node type(s).The process parameterizes (at 1740) the constraints associated with eachnode in the cost vector using selected intensity values and computes (at1750) a cost that is determined from the parameterized constraints forall identified nodes of the cost vector.

As an example, FIG. 18 illustrates computing the cost vector forinternal nodes that are fixed to a second exposure in accordance withthe process 1700 of some embodiments. Accordingly, all nodes 1810corresponding to the node type for internal nodes fixed to the secondexposure are identified. The parameterized constraints for each node areillustrated in the enumerated cost vector 1820. The constraints areparameterized using intensity values corresponding to each node in thevector. Each value can then be summed to compute the cost for the costvector.

III. Optimizing the Cost Function

Some embodiments optimize the cost function to identify an optimalintensity image for each exposure of a multi-exposure photolithographicprocess such that the union of the patterns created/printed from eachexposure intensity image most closely approximates the patterns withinthe region of interest. Some embodiments define the intensity values foreach intensity image (i.e., a particular exposure of a multi-exposurephotolithographic process) through a Fourier expansion. In so doing, theintensities are mapped to a spatial frequency domain allowing someembodiments to bind and minimize the number of feasible solutions in thesolution space. As a result, some embodiments are able to ensure thateach feasible solution represented by the Fourier expansion conformswith the band limitation of a multi-exposure photolithographic printingprocess.

FIG. 19 illustrates a Fourier expansion for two separate intensities1910 and 1920 in accordance with some embodiments. As shown, eachintensity 1910 or 1920 is defined through a summation of different setsof cosine and sine functions. The coefficients a_(mn), b_(mn), c_(mn),and d_(mn) represent the optimization variables for optimizing the costfunction. Specifically, the coefficients are associated with points inthe frequency plane 1930 that are properly band limited. Therefore, whenaltering any one of the coefficients, different intensity values can beexplored for each node within the cost equation while ensuring that theresulting solutions conform with the frequency constraint of theprinting process.

The k_(mn) coefficient of the Fourier expansion represents the spatialfrequency or oscillation as a function of position. As shown, k_(mn) isdefined as2*π*(m/Λ_(x),n/Λ_(y))  (10)

As is known in the art, the Fourier theorem states that any periodicfunction (such as light intensity) can be represented by the sum of aseries of amplitudes and phases of sine waves of one frequency andinteger multiples of that frequency. Accordingly, in the Fourierexpansions 1910 and 1920 of the two intensities I₁ and I₂, Λ_(x)corresponds to the period that the intensity image repeats itself in thex direction and Λ_(y) corresponds to the period that the intensity imagerepeats itself in the y direction. To visualize this, in FIG. 19, K_(x)shows how fast Fourier components oscillate along the x axis. Similarly,K_(y) show how fast Fourier components oscillate along the y axis. Thefurther the distance from the center of the frequency plane 1930 thefaster the oscillations of the component.

This assumption of periodicity, which is necessary for finite Fourieranalysis, causes artificial constraints to be imposed on the intensityimages near the boundary of the repeating image. In order to preventthis, the period of repetition needs to be made larger than the regionof interest, so that opposite edges of the region of interest do notinteract. Alternatively, in the case that the region of interest is apattern that cyclically repeats through the layout or through a portionof the layout, the period of repetition may be made equal to the size ofthe region of interest.

It should be apparent to one of ordinary skill in the art thatdifferently defined frequency planes may be used in conjunction withspecifying the feasible set of intensity values. For example, thecircular shape of the region, Ω, is the largest area of the allowedfrequencies. The actual area, however, can be smaller than the circle,Ω.

FIG. 20 illustrates an example of a region of interest 2005 in someembodiments. Λ_(x) and Λ_(y) correspond to the intensity imageperiodically repeating itself in the x and y directions respectively. Inorder for the repeating images to be out of the region of interest 2005,the region of interest has to be small enough to create buffer areasalong the x and y coordinates, as shown in FIG. 20. In some embodiments,the buffer length (or width) has to be larger than

$\begin{matrix}{C\frac{\lambda}{NA}} & (11)\end{matrix}$where C is a constant, λ is the exposure wavelength and NA is thenumerical aperture and is the refractive index of the optical mediumtimes the sine of the largest angle of incidence on the wafer. In thecase where the region of interest is a pattern that cyclically repeatsthrough the layout or through a portion of the layout, Λ_(x) and Λ_(y)become the dimensions of that cyclically repeating pattern.

In some embodiments, constant C is selected to be larger than or equalto 2 in order to make a large enough buffer area 2010 around the regionof interest 2005 to prevent the repetition of the image caused byFourier analysis to cause artificial constraints to be imposed on theintensity images near the boundary of the region of interest.

By enumerating the coefficients a_(mn), b_(mn), c_(mn), and d_(mn) anintensity image is obtained for the region of interest. The intensityimage represents a feasible exposure that can be obtained based on theconstraints of the photolithographic process. Specifically, theintensity image maps the region of interest to a continuous set ofincreasing and decreasing intensity values which when applied to athreshold filter defines what is printable and the accuracy of theprintability (e.g., does the printed feature have a sufficient width).In other words, an intensity image provides a mathematical simulation ofa feasible exposure that is achievable based on quantifiable constraintsof the photolithographic process.

By parameterizing the cost function with the intensity values, someembodiments determine the similarity between an achievable solution anda desired target specified within the design layout. Therefore, someembodiments must search through the set of feasible solution in order todetermine the best such solutions.

FIG. 21 presents a process 2100 for identifying optimal intensity imagesfor each exposure of a multi-exposure photolithographic process inaccordance with some embodiments. In some embodiments, the process 2100is performed in conjunction with multi-patterning decomposition. Theprocess begins by selecting (at 2110) a solution for the Fourierexpansion from a feasible solution space as described above. Thesolution comprises selecting values for each of the optimizationvariables (e.g., a_(mn), b_(mn), c_(mn), d_(mn)) of the Fourierexpansion. In some embodiments, the optimization variables areinitialized based on what is known about fixed internal nodes.Specifically, it is known that each such node must be printed during aparticular exposure. As such, a starting point is provided from whichthe cost function may be minimized.

The process then uses the solutions to compute (at 2120) intensityvalues for each intensity image. The process uses the computed intensityvalues to specify (at 2130) the cost function by parameterizing the costfunction with intensity values. In so doing, the process realizes a costfor the intensity images. The process then stores (at 2140) the solutionwhen it is necessary. In some embodiments, only solutions that are moreoptimal than all previous iterations will be stored. In some otherembodiments, other solutions are retained from which a determination issubsequently made as to which solution is the best solution.

The process then determines (at 2150) whether to continue iterating. Insome embodiments, the process halts iterations upon an exhaustion ofresources. This may include exceeding a time limit for identifying anoptimal solution. Additionally, this may include identifying a solutionwith a sufficient cost result that satisfies all design constraints ofthe printing process. Accordingly, when the process determines that noadditional iterations are needed, the process returns (at 2160) the bestsolution and the process ends. The returned solution in some embodimentsmight not necessarily be the best possible solution, but only an optimalone given the number of iterations and/or resources expended.

Otherwise, the process selects a new solution to examine by returning tostep 2110 that was described above. In some embodiments, selecting a newsolution includes modifying one or more of the optimization variables(e.g., a_(mn), b_(mn), c_(mn), d_(mn)) in a manner that will minimizethe cost function.

In some embodiments, the optimization iterations are performed inconjunction with or solely using an optimization algorithm such as theLevenberg-Marquardt algorithm. However, it should be apparent to one ofordinary skill in the art that other such optimization algorithms areapplicable to the process 2100 described above to arrive at a set ofoptimal intensity images.

Some embodiments then derive an intensity graph for the region ofinterest from the resulting optimal solution. In some embodiments, thisentails actual generation of the intensity images, an example of whichis illustrated in FIG. 5. In some embodiments, the intensity graphs areused to specify a decomposition for the region of interest by defining acoloring for the patterns appearing within the region of interest such afirst coloring over sections of the patterns identifies polygons to beprinted using a first exposure of a multi-exposure photolithographicprinting process and a second coloring over sections of the patternsidentifies polygons to be printed using a second exposure of amulti-exposure photolithographic printing process. In some embodiments,the decomposition of the polygons is performed by a polygon generator.

FIG. 22 illustrates a process 2200 performed by the polygon generator inaccordance with some embodiments. The process begins when the polygongenerator receives (at 2210) one intensity graph for each exposure of amulti-exposure photolithographic process. The process then imposes (at2220) a grid on the intensities graph. This grid can be the same,coarser, or finer grid than that used to define the region with thenodes.

From the grid, the process identifies (at 2230) the sub-patterns orpolygons used to define a photomask to be used in the photolithographicprinting of the region. Each point of the sub-pattern or polygon isdetermined by comparing the intensity of a node on the grid against athreshold intensity value. Nodes with intensities that exceed thethreshold become part of the sub-pattern or polygon and nodes withintensities lower than the threshold are outside the sub-pattern orpolygon.

The process then defines (at 2240) the photomask to be used in anexposure of a multi-exposure printing process based on the intensity andthreshold determinations that yielded the sub-patterns. The sub-patternsfrom each particular intensity graph define each photomask.Specifically, some embodiments utilize an EBeam process to etch thephotomask with the polygons at the specified grid locations. In somesuch embodiments, the sub-patterns define the shapes to be etched intothe photomask whereas in other embodiments the sub-patterns are mappedto the photomask using various optical proximity correction techniquesand/or sub-resolution assist features (e.g., scatter bars, OAI, etc.).

IV. Software Architecture Overview

FIG. 23 presents a conceptual diagram of the software architecture 2300for implementing some embodiments of the invention. As shown, thearchitecture 2300 processes a design layout 2310 using: (1) a regionselector 2320, (2) a node designator 2330, (3) a constraints database2335, (4) a costing engine 2340, (5) an optimization engine 2350, and(6) a polygon generator 2360.

In some embodiments, the region selector 2320 indentifies the particularregions of interest within the design layout 2305. In some embodiments,the region selector 2320 operates in conjunction with a globaloptimization module. In some such embodiments, the global optimizationmodule may be used to initially decompose various regions of the designlayout. For those complex regions of the design layout 2310 that theglobal optimization module cannot optimize, the global optimizationmodule marks such regions and defers processing of the marked regionssuch that the marked region are processed in accordance with someembodiments of the invention. Accordingly, the global optimizationmodule is not necessary to the operation of some embodiments.

In other words, the double patterning optimization technique of someembodiments is scalable to perform over particular regions of interestin a layer of a layout design, entire layers of the layout design, orover the entire layout design. For instance, FIG. 24 illustrates aregion of interest 2410 that has been selected for multi-exposureintensity optimization as encompassing an entire design layout or layerof the design layout. FIG. 25 illustrates a region of interest 2510 thathas been selected for multi-exposure intensity optimization asencompassing only a spatial sub-region of a layer of a design layout2520.

Once the region of interest has been identified by the region selector2310, the node designator 2330 defines the nodes over the region ofinterest. In some embodiments, the costing engine 2340 assigns variousconstraints from the constraints database 2335 to the nodes based on thenode types. The costing engine 2340 also defines the cost function forthe region of interest. The optimization engine 2350 then performs theoptimization of the cost function in order to identify a set ofintensities that best print the design layout in a multi-exposurephotolithographic printing process. The resulting set of optimizedintensities are then processed by the polygon generator 2360 to definethe photomasks used in decomposing the region of interest for themulti-exposure photolithographic printing process.

V. System Architecture Overview

Many of the above-described processes and engines are implemented assoftware processes that are specified as a set of instructions recordedon a machine readable medium (also referred to as computer readablemedium). When these instructions are executed by one or morecomputational element(s) (such as processors or other computationalelements like ASICs and FPGAs), they cause the computational element(s)to perform the actions indicated in the instructions. Computer is meantin its broadest sense, and can include any electronic device with aprocessor. Examples of computer readable media include, but are notlimited to, CD-ROMs, flash drives, RAM chips, hard drives, EPROMs, etc.

In this specification, the term “software” is meant in its broadestsense. It can include firmware residing in read-only memory orapplications stored in magnetic storage which can be read into memoryfor processing by a processor. Also, in some embodiments, multiplesoftware inventions can be implemented as sub-parts of a larger programwhile remaining distinct software inventions. In some embodiments,multiple software inventions can also be implemented as separateprograms. Finally, any combination of separate programs that togetherimplement a software invention described here is within the scope of theinvention.

FIG. 26 is a block diagram of an illustrative computing system 2600suitable for implementing an embodiment of the present invention.Computer system 2600 includes a bus 2606 or other communicationmechanism for communicating information, which interconnects subsystemsand devices, such as processor 2607, system memory 2608 (e.g., RAM),static storage device 2609 (e.g., ROM), disk drive 2610 (e.g., magneticor optical), communication interface 2614 (e.g., wireless 802.11b/g orEthernet card), input device 2612 (e.g., keyboard or cursor control),and output device 2620 (e.g., display monitor).

According to one embodiment, computer system 2600 performs specificoperations by processor 2607 executing one or more sequences of one ormore instructions contained in system memory 2608. Such instructions maybe read into system memory 2608 from another computer readable/usablemedium, such as static storage device 2609 or disk drive 2610. Inalternative embodiments, hard-wired circuitry may be used in place of orin combination with software instructions to implement the invention.Thus, embodiments of the invention are not limited to any specificcombination of hardware circuitry and/or software. In one embodiment,the term “logic” shall mean any combination of software or hardware thatis used to implement all or part of the invention.

The term “computer readable medium”, “computer readable storage medium”,or “computer usable medium” as used herein refers to any tangible mediumthat participates in providing instructions to processor 2607 forexecution. Such a medium may take many forms, including but not limitedto, non-volatile media and volatile media. Non-volatile media includes,for example, optical or magnetic disks, such as disk drive 2610.Volatile media includes dynamic memory, such as system memory 2608.Common forms of computer readable media includes, for example, floppydisk, flexible disk, hard disk, magnetic tape, any other magneticmedium, DVD-ROM, DVD-RAM, CD-ROM, any other optical medium, RAM, PROM,EPROM, FLASH-EPROM, any other memory chip or cartridge, or similartangible medium from which a computer can read.

In an embodiment of the invention, execution of the sequences ofinstructions to practice the invention is performed by a single computersystem 2600. According to other embodiments of the invention, two ormore computer systems 2600 coupled by the communication interface 2614(e.g., LAN, PTSN, or wireless network) may perform the sequence ofinstructions required to practice the invention in coordination with oneanother.

Computer system 2600 may transmit and receive messages, data, andinstructions, including program, i.e., application code, through thecommunication interface 2614. Received program code may be executed byprocessor 2607 as it is received, and/or stored in disk drive 2610, orother non-volatile storage for later execution.

While the invention has been described with reference to numerousspecific details, one of ordinary skill in the art will recognize thatthe invention can be embodied in other specific forms without departingfrom the spirit of the invention. Thus, one of ordinary skill in the artwould understand that the invention is not to be limited by theforegoing illustrative details, but rather is to be defined by theappended claims.

What is claimed is:
 1. A method for decomposing a particular pattern ona layer of a design layout into multiple sub-patterns for multiplephotolithographic exposure operations, the layer comprising a pluralityof regions, the method comprising: receiving a region of interestcomprising the particular pattern defined by a set of edges; identifyingcorner edges where a plurality of edge segments from the set of edgesare joined together; processing the set of edges by rounding theidentified corner edges of the particular pattern; demarcating the setof processed edges by defining a set of nodes that follows contours ofthe set of processed edges, the set of nodes comprising a set ofinternal edge nodes defined inside the particular pattern and a set ofexternal edge nodes defined outside the particular pattern, the spacebetween the internal edge nodes and external edge nodes specifying anacceptable edge placement error during pattern decomposition; specifyingintensity values for the set of internal nodes and the set of externalnodes, wherein the intensity value for the set of internal nodes isdifferent than the intensity value for the set of external nodes; and bya computer, searching, based on the specified intensity values, asolution space of lithographically feasible images to identify at leasttwo images for decomposing the particular pattern such that a union ofthe images closely approximates the particular pattern.
 2. The method ofclaim 1, wherein the corner edges are rounded to satisfy a set ofphysical constraints associated with shapes of the images in thesolution space.
 3. The method of claim 1, wherein the internal edgenodes are defined over a first portion of the particular pattern, themethod further comprising defining a second set of internal nodes over asecond remaining portion of the particular pattern not covered by theset of internal edge nodes.
 4. The method of claim 3, wherein a numberof defined internal nodes has to exceed a maximum frequency of a bandlimited signal used during pattern decomposition by at least a factor oftwo in order to represent all features within the region of interestwith sufficient granularity.
 5. The method of claim 4, wherein themaximum frequency of the band limited signal is based on resourcesavailable to process the region of interest.
 6. The method of claim 1,wherein the particular pattern is part of a larger pattern that extendsbeyond the region of interest, the method further comprising:identifying a set of fixed nodes located in the particular pattern nextto a boundary of the region of interest where the particular patternextends beyond the region of interest; and assigning an intensity valueto the set of fixed nodes.
 7. The method of claim 6, wherein theinternal nodes that are not identified as fixed nodes are assigned anintensity value based on the set of fixed nodes in order to minimize anumber of intensity transitions that occur when printing the particularpattern.
 8. The method of claim 6, wherein the fixed nodes are printedat a same exposure as a portion of the particular pattern that extendsbeyond the boundary of the region of interest during a photolithographicexposure operation.
 9. The method of claim 6, wherein a set of designconstraints are applied to the fixed nodes and a different set of designconstraints are applied to the internal nodes that are not identified asfixed nodes.
 10. The method of claim 6, wherein the set of fixed nodesis a first set of fixed nodes, the method further comprising:identifying a second set of fixed nodes located in the particularpattern next to a different boundary of the region of interest where theparticular pattern extends beyond the region of interest; and assigningan intensity value to the second set of fixed nodes, wherein theintensity value assigned to the first set of fixed nodes is differentthan the intensity value assigned to the second set of fixed nodes. 11.A non-transitory computer readable medium storing a computer program forexecution by at least one processing unit, the computer program fordecomposing a particular pattern on a layer of a design layout intomultiple sub-patterns for multiple photolithographic exposureoperations, the layer comprising a plurality of regions, the computerprogram comprising sets of instructions for: receiving a region ofinterest comprising the particular pattern defined by a set of edges;identifying corner edges where a plurality of edge segments from the setof edges are joined together; processing the set of edges by roundingthe identified corner edges of the particular pattern; demarcating theset of processed edges by defining a set of nodes that follows contoursof the set of processed edges, the set of nodes comprising a set ofinternal edge nodes defined inside the particular pattern and a set ofexternal edge nodes defined outside the particular pattern, the spacebetween the internal edge nodes and external edge nodes specifying anacceptable edge placement error during pattern decomposition; specifyingintensity values for the set of internal nodes and the set of externalnodes, wherein the intensity value for the set of internal nodes isdifferent than the intensity value for the set of external nodes; andsearching, based on the specified intensity values, a solution space oflithographically feasible images to identify at least two images fordecomposing the particular pattern such that a union of the imagesclosely approximates the particular pattern.
 12. The non-transitorycomputer readable medium of claim 11, wherein the set of instructionsfor defining the external edge nodes comprises a set of instructions fordefining the external edge nodes over a first portion of the region ofinterest not comprising the particular pattern, the computer programfurther comprising a set of instructions for defining a second set ofexternal nodes over a second remaining portion of the region of interestnot comprising the particular pattern and not covered by the set ofexternal edge nodes.
 13. The non-transitory computer readable medium ofclaim 12, wherein the computer program further comprises a set ofinstructions for defining a particular threshold intensity, whereinintensity values below the particular threshold are designated as notfor printing and intensity values above the particular threshold aredesignated for printing.
 14. The non-transitory computer readable mediumof claim 13, wherein the plurality of external nodes are assigned anintensity value that is below the particular threshold intensity. 15.The non-transitory computer readable medium of claim 13, wherein theprogram further comprises a set of instructions for defining a pluralityof internal nodes over the particular pattern, wherein each internalnode is assigned an intensity value that is above the particularthreshold intensity.
 16. The non-transitory computer readable medium ofclaim 11, wherein the particular pattern is part of a larger patternthat extends beyond the region of interest, the computer program furthercomprising sets of instructions for: identifying first and second setsof fixed nodes located in the particular pattern next to a boundary ofthe region of interest where the particular pattern extends beyond theregion of interest, wherein the first set of fixed nodes is located at adifferent boundary than the second set of fixed nodes; and assigning afirst intensity value to the first set of fixed nodes and a seconddifferent intensity value to the second set of fixed nodes.
 17. A systemcomprising: a processor for executing sets of instructions; and a memoryfor storing a computer program for decomposing a particular pattern on alayer of a design layout into multiple sub-patterns for multiplephotolithographic exposure operations, the layer comprising a pluralityof regions, the computer program comprising sets of instructions for:receiving a region of interest comprising the particular pattern definedby a set of edges; identifying corner edges where a plurality of edgesegments from the set of edges are joined together; processing the setof edges by rounding the identified corner edges of the particularpattern; demarcating the set of processed edges by defining a set ofnodes that follows contours of the set of processed edges, the set ofnodes comprising a set of internal edge nodes defined inside theparticular pattern and a set of external edge nodes defined outside theparticular pattern, the space between the internal edge nodes andexternal edge nodes specifying an acceptable edge placement error duringpattern decomposition; specifying intensity values for the set ofinternal nodes and the set of external nodes, wherein the intensityvalue for the set of internal nodes is different than the intensityvalue for the set of external nodes; and searching, based on thespecified intensity values, a solution space of lithographicallyfeasible images to identify at least two images for decomposing theparticular pattern such that a union of the images closely approximatesthe particular pattern.
 18. The system of claim 17, wherein the set ofinstructions for defining the internal edge nodes comprises a set ofinstructions for defining the internal edge nodes over a first portionof the particular pattern, the computer program further comprising a setof instructions for defining a second set of internal nodes over asecond remaining portion of the particular pattern not covered by theset of internal edge nodes.
 19. The system of claim 18, wherein a numberof defined internal nodes has to exceed a maximum frequency of a bandlimited signal used during pattern decomposition by at least a factor oftwo in order to represent all features within the region of interestwith sufficient granularity.
 20. The system of claim 17, wherein the setof instructions for defining the external edge nodes comprises a set ofinstructions for defining the external edge nodes over a first portionof the region of interest not comprising the particular pattern, thecomputer program further comprising a set of instructions for defining asecond set of external nodes over a second remaining portion of theregion of interest not comprising the particular pattern and not coveredby the set of external edge nodes.
 21. The system of claim 17, whereinthe particular pattern is part of a larger pattern that extends beyondthe region of interest, the computer program further comprising sets ofinstructions for: identifying first and second sets of fixed nodeslocated in the particular pattern next to a boundary of the region ofinterest where the particular pattern extends beyond the region ofinterest, wherein the first set of fixed nodes is located at a differentboundary than the second set of fixed nodes; and assigning a firstintensity value to the first set of fixed nodes and a second differentintensity value to the second set of fixed nodes.